In bridge, how much support do you need to raise a responder?
This is a matter of partnership understanding and taste more than systemic; most natural bidding systems may be played either way.

First, you would never make strong raises of partner's major (splinter, jump to 3M or 4M, etc) with only 3-card support. This is exclusively for the auction 1m-1M-2M. 4-3 fits are much less fun to play at the 4-level than the 2-level.

Second, the reason that you're bidding this way is that you expect that 2M will play better than 1NT. Many partnerships will only raise on three with shortness in an unbid suit. Very few partnerships will raise with a 4333 hand. Remember, ruffing values in the short hand are good; if you're forced to ruff in the long hand in a 4-3 fit, you may find yourself losing trump control. On this note, I don't like Qxx as a minimum; if you have shortness, 3 small trumps are probably fine because some of them will be winners.

Third, remember that partner may have enough to invite or bid a game after your limiting bid. If partner doesn't expect you to raise on 3, they may bid inaccurately. Make sure that their second bid differentiates between a 4-card suit and a 5-card suit. A common agreement is that 1m-1M-2M-3M is invitational with 5+ in the major, while 1m-1M-2M-2NT is invitational with 4; similarly 3NT by responder asks opener to make a choice of games. There are conventional treatments that are probably better: this article on bridgewinners describes the spiral raises convention.