Number which is simultaneously sum of 2 and 3 squares

Obviously, if there are any solutions to $a^2 =b^2 + c^2$, then you can add $x^2$ to both sides and get your $m$'s.

For example, $25 + x^2 = 9 + 16 + x^2$ for all natural numbers $x$.

You can even have $m$ be a perfect square AND a sum of two squares AND a sum of three squares (take $x=12$ above).

There is a possibly famous identity: $$\Large 10^2+11^2+12^2=13^2+14^2$$ which meets your criteria.

Here $m$ is $365$.

Alternatively you could use a formula to generate examples; for example

$$m=\left(a^2+b^2+c^2\right)^2+\left(a^2-c^2\right)^2=\left(a^2+c^2\right)^2+\left(2cb\right)^2+\left(a^2+b^2-c^2\right)^2$$

originating from the identity $$\left(a^2+b^2+c^2\right)^2-\left(a^2+b^2-c^2\right)^2=\left(2ca\right)^2+\left(2cb\right)^2$$ and using $$\left(a^2+c^2\right)^2-\left(a^2-c^2\right)^2=\left(2ca\right)^2$$

to substitute for $\left(2ca\right)^2$