For each fixed $c$, the equation $a+b=22-2c$ has $22-2c+1$ non-negative integer solutions (since $a$ runs from $0$ to $22-2c$).

$c$ ranges from $0$ to $11$ (so that $22-2c\ge 0$). The counts are:

$c=0$: 23,   $c=1$: 21,   $c=2$: 19,   ...,   $c=11$: 1.

Sum of the arithmetic series $1+3+5+\dots +23$ (12 odd terms):

$\sum =12^2=144$.