Pell equations are relevant to certain sequences, recurrence relations. E.g., see this answer. I'll use a formula from the answer with $x_1,y_1,D$.
Sometimes a Pell equation doesn't seem to exist for a certain sequence, as seen in this case and this case (see the ends of the answers).
In this case, a Pell equation does exist.
Let $a_n=\frac{(2+\sqrt{3})^n+(2-\sqrt{3})^n}{2}$, $n\ge 0$, $n\in\mathbb Z$.
Then $b_n=2a_n$ ($n\ge 0$, $n\in\mathbb Z$) is the sequence of integers (you wanted to prove they're integers) you want.
$a_0=1$, $a_1=2$.
We could continue for curiosity, but it's not needed:
$a_2=7$, $a_3=26$, $a_4=97$, etc.
$x_1=2$, $y_1=1$, $D=3$.
$2^2-3\cdot 1^2=1$. The equality holds!
Therefore we can expect a Pell equation.
And indeed, $\frac{a_n^2-1}{3}$ with $n\ge 0$, $n\in\mathbb Z$ is always a perfect square.