There is the following sequence of positive solutions (according to Mathematica): $a_n=\frac{1}{8} \left(\left(\sqrt{6}-2\right) \left(2 \sqrt{6}+5\right)^n-\left(\sqrt{6}+2\right) \left(5-2 \sqrt{6}\right)^n-4\right),$ and $b_n=\frac{1}{12} \left(-\left(\sqrt{6}-3\right) \left(2 \sqrt{6}+5\right)^n+\left(\sqrt{6}+3\right) \left(5-2 \sqrt{6}\right)^n-6\right)$.

There is the following sequence of positive solutions (according to Mathematica): $a_n=\frac{1}{8} \left(\left(\sqrt{6}-2\right) \left(2 \sqrt{6}+5\right)^n-\left(\sqrt{6}+2\right) \left(5-2 \sqrt{6}\right)^n-4\right),$ and $b_n=\frac{1}{12} \left(-\left(\sqrt{6}-3\right) \left(2 \sqrt{6}+5\right)^n+\left(\sqrt{6}+3\right) \left(5-2 \sqrt{6}\right)^n-6\right)$.

Your Diophantine equation is known as the Pell equation.