No, tput won't do this (normally) because you're likely to use only the predefined/standard terminal capabilities, which don't have that combination.
With ncurses, you could define your own terminal description with a user-defined capability, and tput would work with that.
Something like this would combine colors:
infocmp -x >foo
printf '\tfgbg=\\E[3%p1%;4%p2%dm,\n' >>foo
sudo tic -x foo
Then (the example is for 8-colors)
tput fgbg 4 0
would set the foreground to blue and background to black. However, there are a lot of possibilities. Using the predefined capabilities involves less effort.