点$A\left(m,1\right)$关于$y$轴的对称点坐标为$\left(-m,1\right)$.
$\because$点$\left(-m,1\right)$在一次函数$y=3x+4$的图象上，
$\therefore 1=-3m+4$，
$\therefore m=1$.
故答案为：$1$.

点$A\left(m,1\right)$关于$y$轴的对称点坐标为$\left(-m,1\right)$.
$\because$点$\left(-m,1\right)$在一次函数$y=3x+4$的图象上，
$\therefore 1=-3m+4$，
$\therefore m=1$.
故答案为：$1$.