Honestly, depends on hole card, whether you're holding a set or 2 pairs since turn card is a variable factor that changes the calculation, very different from other cards such as straight draw and flush draw. But i'll give you 3 situations. If you want full details of how to calculate them, read the whole comment. If you just wanna get the answer, go down to bottom of my comment.
If your hole cards didn't pair the board, then chance of board pairing is
9 outs to pair the board from the flop
19.15% chance of pairing by the turn since it's 9 cards out of 47, which is 0.1915 = 19.15%
From turn, your outs increase, so it's 12 out of 46, which is 26.09%
From the flop, if you're expecting to board by river, then you simple get chance of not pairing board from flop-turn and turn-river and multiply it. Since chance of not pairing by turn from flop is 100-19.15=80.85% and turn to river is 100-26.09=73.91%, you multiply them(0.8085 x 0.7391 = 0.5976 = 56%), which gives percentage of not pairing the board from flop to river. Therefore, your chance of pairing the board if you didn't pair the flop and to see if board will pair by the river is 40.24%
If you have a set, then using this precise out calculation using your calculator, you do the same thing but adjust number of outs.
From flop - turn
There are 7 outs. so simply 7/47 = 14.89%
From turn - river
Your outs increase to 10. So simply 10/46 = 21.74%
Get percentage of not coming out and multiply them together to get from flop-river
(0.8511 x 0.7826 = 0.6661 = 66.61%) Since chance of not pairing when holding a set is 66.61%, your chance of pairing the board from flop to river when holding a set, is 33.39%.
If you flopped 2 pairs and seeking full house, then there's no adjustment in outs. You're looking for one of the 4 cards to come out by the river. So simply, from flop to turn, you calculate 4/47, which is 0.051 = 8.51%. From turn to river is 4/46, which is 0.086957 = 8.70%. To calculate making ur full house from 2 pairs, simple calculate chance of not coming out from flop-turn and turn-river and then subtract that from 100 to get ur chance. So it's (0.9149 x 0.9130) = 0.8353 = 83.53%. Subtract that from 100 to get % of making ur full house from flop to river, which is 0.1647 = 16.47%
Prolly got tired of reading all my long, crappy comment? I'll digest it down to simplify and summarize what I wrote.
If your hole cards didnt pair the board, then chance of board pairing is
Flop to turn = 19.15%
Turn to river = 26.09%
Flop to river = 40.24%
If you have a set, then chance of pairing the board is
Flop to turn = 14.89%
Turn to river = 21.74%
Flop to river = 33.39%
If you flopped 2 pairs, chance of making your full house is
Flop to turn = 8.51%
Turn to river = 8.70%
Flop to river = 16.47%
I hope this gives full details about pairing the board and chance of making ur full house/quads.