If I have an opening strength hand and my best suit is the same major as the opening opposition's suit bid, I don't think I can cue bid in this event. How can I tell my partner that we want to play in this suit?
Are you sure you want to play in that suit? Especially if your opponents open a major only if they hold a 5-card, you already know where 10 of the trumps are. Chances of a fit with your partner are slim and the length of that opponent can really hurt you.
Consider remaining silent for the moment and double when they reach a manche. If you are really bent on playing and you also hold something in 2, 3 or all of the other suits, consider bidding 1NT.
If you would like to "play" (that is, defend) in a suit that your opponents have bid, you make what is called a "trap pass." You know that you want to play in that suit, but your opponent doesn't know that for now. In essence, you have set a trap for him. Because you are doing this, you can't afford to tell your partner anything.
If you and opener are both "long and strong" in the suit, your partner is likely to be very weak. Realize that opener probably has all the missing honors that you don't have, while your partner probably has a singleton, if not a void in the suit. Assuming a reasonable level of strength in the three other suits, s/he will double the opener's bid for "takeout." Then you can pass for penalties and make the opponents play in "your" suit for more than half the tricks when you and partner have more than half the points.
This penalty pass means that "I am at least as strong in the suit as opener, maybe opener and dummy put together." This implies strength of at least something like AKxxx, or QJTxx. You're telling partner, "We should lead this suit, knock out dummy's trumps, and then play a quasi no trump game in your (the doubling partner's) three strong suits."
You might want to play in that suit, but your partner is probably void and would really prefer some other suit. You've got two options here:
1) Sit in the bushes. Underbid your hand a bit, see if the opposition is willing to bid a serious contract. Double them at the last minute for a nice down score!
2) Bid your next-best suit. If there's a lot of points in this deck, then your partner might bid the 3rd suit and then you know you've got all the suits covered and you can play no trump. Or, partner might like your next-best suit, and you'll go on to some small game there.
When your opponent has your suit, that really just means that neither of you is going to be able to play that suit very well.
A lot depends on what the other bid is likely to mean, and whether other people bid beforehand (question doesn't say). For example 1H-? is different from 1D-1NT-2H-? or 1D-1H-DBL-PASS-? each of which might have a heart 4 or 5 card suit which otherwise could be bid.
You might also want to consider whether to double to show values, pass and wait (you can always re enter the bidding next round which will say a lot to your partner), or try and find a different fit which allows a cross ruff in hearts and some suit you are short in (once trumps are eliminated in the other short-suited hand).
But sometimes you just don't have a bid, and its fine to let things pan out and the other side try to make it. After all, if they stick to hearts or NT you have potential defensive strength they won't expect (although perhaps entry issues if weak elsewhere) and if they change suit, pass at low level, or panic when doubled at a higher level, you can always re-enter the bidding at a better time.