Given the quadratic equation $x^{2}-5x+3=0$, we know its roots are $m$ and $n$. To find $m^{2}+n^{2}$, we can use the sum and product of roots formulas for quadratic equations.

1. The sum of roots formula is $m+n=-\frac{b}{a}$. For our equation, $a=1$ and $b=-5$, so we have:
$$m+n=-\frac{-5}{1}=5$$

2. The product of roots formula is $mn=\frac{c}{a}$. For our equation, $a=1$ and $c=3$, so we have:
$$mn=\frac{3}{1}=3$$

3. To find $m^{2}+n^{2}$, we use the identity $(m+n)^{2}=m^{2}+2mn+n^{2}$. Rearranging for $m^{2}+n^{2}$ gives us:
$$m^{2}+n^{2}=(m+n)^{2}-2mn$$

4. Substituting the values of $m+n$ and $mn$ into the equation:
$$m^{2}+n^{2}=5^{2}-2\times 3=25-6=19$$

Therefore, the sum of the squares of the roots of the equation $x^{2}-5x+3=0$ is $\boxed{19}$.