This is a special case of IMO Shortlist 2010 N2, with $m = 3x$ and $n = 2y + 1$.

We have $9x^2+2\cdot3^{2y+1}=3x(2\cdot2^{2y+1}-1)$. Setting $a=3x$ and $b=2y+1$ gives the equation: $a^2+2\cdot3^b=a(2\cdot2^b-1)$. We then do casework in b. Case 1: $b=1$ We have $a^2-3a+6=0$, which has no real roots. Case 2: $b=3$ We have $a^2-15a+54=0$, which has roots $a=6, 9$. This gives the tuples $(6, 1)$ and $(9, 1)$. Case 3: $b=5$ We have $a^2-63a+486=0$, which has roots $a=9, 54$. This gives the tuples $(9, 2)$ and $(54, 2)$. Case 4: $b\ge7$, b odd