Before I start, I'll confirm your point regarding lines which end "/fold." We will never, in a million years, fold this hand. If we bet, it's to call. If we check, it's not to fold. So yes, with those options gone, we have three left, one of which I basically won't use (check/call) because I tend not to favour such passive lines. I'm sure there are hypothetical villains against whom this may be preferable, but they are very rare so it's not a line I will be advocating. What I'll now do is discuss three types of typical villain and how I'd approach this hand versus each:

Loose/Aggressive:

Against these guys, this is an incredibly easy check/shove. Their range isn't very strong at all by this point (he would have raised the majority of his best made hands - almost certainly anything QT w/ a redraw or better - on the flop) so we are looking to push an equity edge by putting money in against him. By checking and almost certainly inducing a barrel, we are able to pick up a decently large pot w/o showdown by getting him to stab wide and then fold his equity (he'll usually have a fair chunk with us only holding a pair). Alternatively, he gets it in which is fine because it'll be very uncommon that we're in bad shape. A lot of the hands he'll have here to bet/call will be pair + draws which we clearly destroy. If they do check behind it's also not a disaster as we'll get to see the river and can probably play ok (we don't have a hand that massively requires protection and villain probably won't check a hand with tons of outs) - certainly not what we're hoping for though.

Loose/Passive:

More passive villains I would want to lead against. Again, I think we're ahead of their ranges, although perhaps less so than versus the aggro guys (they will have flatted more made hands on the flop). However, I'd be very sad if I didn't give them the chance to put more money in w/ a diamond draw or QJ9 etc. It's also nice that we should be able to play river very well, even if we're potentially forced to give up the lead on a blank, as they are unlikely to bluff us. Against the LAGs, we might fire here, miss river and still want to show it down which will often be very expensive. Against the passive guys, we can check and they'll usually check behind everything that we beat as well as a decent chunk of the stuff which beats us meaning we should get to realise our equity on a lead more completely than versus the aggros. We also won't get the same value from stab/folds which we'd have gotten from the aggro guys so, as is usually the case versus passive players, we're better off doing the betting for them.

Tight/Aggressive:

I think these guys are also a check/shove. My reasoning for this is that a lot will, as with their looser counterparts, have raised many of their stronger made hands OTF leaving them with a somewhat weak range. However, unlike those guys, they may have been less comfortable raise/getting it in with some of their bigger draws as they didn't have that many nut outs (with us holding nut diamonds, it's very hard for them to have a draw which performs well versus our flop getting it in range). However, once we check, they are very likely to fire pretty much anything they have which is good for us as we can then move in over the top (most of their range will want to bet versus our OOP check). If we're called, it'll depend on how they interpret that line as to exactly how well we'll be doing (some will think it looks like AA + diamonds etc. and call off two pairs, others will see QQ and potentially only call off draws with a lot of outs which is amazing for us). What we can basically guarantee though is that we'll be doing well enough to make money as they'll either be getting it in bad too much or bet/folding enough that it doesn't matter, especially given we still have strong equity versus most anything. The fact that this line could lead to them making huge errors in folding two pair is particularly attractive to my way of thinking (I want to give people the chance to make a mistake).

Final Thoughts:

The last thing I do wish to note is that I have clearly assumed something here and that is that villain won't have a range which consists primarily of strong made hands. This is not to say that sets and decent QTs won't show up sometimes, I just believe that it's somewhat unlikely that those hands show up with regularity, especially from an aggressive opponent with position (less scared of raising and seeing ugly turns as they can check them back quite frequently). We also hold a Q to block both top set and top two whilst most every wrap card is free (we block a single 9 as well as a K and an A although those two overcards feature in far fewer big draws than the Js, 9s and to a slightly lesser extent 8s).

I also want to point out that some villains won't fit neatly into the above categories - they may tread the line between loose and tight or between passive and aggressive etc. In these cases, it's up to the judgment of Hero to work out how one expects a given villain to play certain hands against our checks and our bets. My personal belief is that there are two major factors which should come into play: how wide we expect them to barrel against a check (the wider, the happier we are to check/shove), and do we expect them to bluff rivers if we bet turn and then check to them (the more likely they are to do that, equally, the more we should lean towards a check/shove). Happily, a propensity towards both those actions is usually found in the same villains which helps make our lives easier as many will often be either a clear bet/call or a clear check/shove.

I think this hand isa great spot for the c/r line against any somewhat aggressive reg, we take so much money from light flop peels that stab turn, and we aren't really unhappy when it goes check/check. 

I think this situation more interesting when we have  a similar strength hand but without the significant showdown value that the Q gives us on a lot of rivers where we don't hit our flush or straight. If we swap the Q for a 7 or maybe even a J, I think there is still a strong case to be made for a c/r; however unlike when we have the Q, when the turn checks through and the river bricks I'm hating life. We have so little showdown value that I feel like we should be bluffing but at the same time villain probably expects our range to be very capped due to the turn check and it's hard for us to rep many made hands that would value bet.

I don't remember the hand... I'm not villain am I? 

When I say that the Q gives us showdown value I mean that; when the turn goes check/check and the river goes check/check we are going to win a decent amount (although yes I do think there is potential for some c/c rivers).

If you change the Q, however it sucks when we miss our turn c/r give up river and villain checks back some random pair that he wouldn't have even considered a call with. 

I'm barreling here like 80% of the time at least. I'd need a solid read that villain bets pretty mindlessly when checked to if planning to go for a check-raise. Most reasonably competent players should get suspicious when we all of a sudden check the brick turn after c-betting a very coordinated flop into two players, especially after opening UTG.