This problem was handled by Hooley and Guo (independently). See 

Hooley - On ternary quadratic forms that represent zero.

Guo - On solvability of ternary quadratic forms.

Hooley obtained the (sharp) lower bound of the form $H^2/(\log H)^{3/2}$ for the corresponding counting problem. Guo obtained an asymptotic formula for the slightly different problem where $a$ and $b$ are assumed to be square-free (i.e. the numerators and denominators are square-free).

An asymptotic formula without the square-free assumption does not seem to be known.