Count integer pairs $(x,y)$ with $4x^2+y^2<52$. Only $|x|\le 3$ works since $4(4{)}^2=64>52$.

For each $x$, count integer $y$ with $y^2<52-4x^2$:

$x=0$: $y^2<52\Rightarrow y\in \{-7,\dots ,7\}$, giving 15 values.

$x=\pm 1$: $y^2<48\Rightarrow |y|\le 6$, giving 13 values each, total $2\times 13=26$.

$x=\pm 2$: $y^2<36\Rightarrow |y|\le 5$, giving 11 values each, total $2\times 11=22$.

$x=\pm 3$: $y^2<16\Rightarrow |y|\le 3$, giving 7 values each, total $2\times 7=14$.

Total $=15+26+22+14=77$.