¿How can I get the implied vol from a swaption when I have a vol surface with the maturity of the option and the tenor of the swap? For example I want to know what is the volatility for a swaption that gives the holder the right to pay 6.2% in a 3-year swap starting in 5 years.
Swaption vol can have 3 dimensions: option expiry, underlying tenor and strike.
In your example, if nothing is said, then it's probably ATM (at the money) volatility which means it's the vol for a Swaption with a strike equal to the forward of the underlying.
So if you only have a surface, and not a cube, you probably don't have exactly the information you need.
If you do have vol data for multiple strikes, even if you don't have the 6.2% strike, you could fit SABR parameters to the smile to determine the vol for the strike you need.
Alternatively, if you have a cap surface and a Swaption ATM surface, you could also build other Swaption strike vols with a method called lifting from caps.