Assume that $X$ and $Y$ are independent with known means/variances.
Unfortunately, I don't think there's a nice standard form for the distribution of $Z$, but I'd be happy to be shown otherwise.
Let $X' = \frac{X - a}{\sqrt{v_1}} \sim \mathcal{N}(\frac{m_1 - a}{\sqrt{v_1}}, 1)$ and $Y' = \frac{Y - b}{\sqrt{v_1}} \sim \mathcal{N}(\frac{m_2 - b}{\sqrt{v_2}}, 1)$.
The square of each is noncentral $\chi^2$ with parameters $k = 1$, $\lambda_1 = \frac{(m_1 - a)^2}{v_1}$ (for $X'$), $\lambda_2 = \frac{(m_2 - b)^2}{v_1}$ (for $Y'$).
We can then write the pdf of their linear combination $Z^2 = v_1 X'^2 + v_2 Y'^2$ with a Laguerre expansion (Castaño-Martínez and López-Blázquez, 2005, (3.2)):
$$
f(z) =
\frac{1}{2 \beta} e^{-\frac{z}{2\beta}}
\sum_{k \ge 0}
\frac{k! c_k}{(1)_k}
L_k\left( \frac{2 z}{4 \beta \mu_0} \right)
$$
where $\mu_0 > 0$ and $\beta > 0$ are arbitrary parameters,
$L_k$ is the $k$th Laguerre polynomial,
I think $(1)_k$ is the rising factorial so that it's $k!$ and cancels with the $k!$ in the numerator,
and the $c_k$ satisfy these recurrences:
$$
c_0 =
\frac{1}{\mu_0}
\exp\left( - \frac{1}{2} \sum_{i=1}^2 \frac{\lambda_i}{1 + \frac{\beta \mu_0}{v_i (1 - \mu_0)}} \right)
\prod_{i=1}^2 \left( 1 + \tfrac{v_i}{\beta} \left(\tfrac{1}{\mu_0} - 1\right) \right)^{-1/2}
\\
c_k = \frac{1}{k} \sum_{j=0}^{k-1} c_j d_{k-j}
\\
d_j =
- \frac{j \beta}{2 \mu_0}
\sum_{i=1}^2
\lambda_i v_i (\beta - v_i)^{j-1}
\left( \frac{\mu_0}{\beta \mu_0 + v_i(1 - \mu_0)} \right)^{j+1}
+ \sum_{i=1}^2 \tfrac12 \left(\frac{1 - v_i/\beta}{1 + (v_i/\beta) (1/\mu_0 - 1)}\right)^j
$$
In practice, you can truncate the sum after a few values of $k$. The authors show an error bound in (3.9), though according to Bausch (2013) (who gives a more computationally efficient approximation for linear combinations of central chi-squareds) the bound is quite conservative.
If you want the cdf there's a similar expression (3.5):
$$
F(z) =
\frac{1}{(2 \beta)^2} z e^{-\frac{z}{2\beta}}
\sum_{k \ge 0}
\frac{k! m_k}{(2)_k}
L_k^{(1)} \left( \frac{z}{\beta \mu_0} \right)
$$
where again I think $(2)_k = (k+1)!$, $L_k^{(1)}$ is a generalized Laguerre polynomial, and
$$
m_0 = 8 \exp\left( -\tfrac12 \sum_{i=1}^2 \frac{\lambda_i}{1 + \frac{\beta \mu_0}{v_i (1 - \mu_0)}} \right) \frac{\beta^2}{1-\mu_0} \prod_{i=1}^2 \left( \beta \mu_0 + v_i (1 - \mu_0) \right)^{-1/2}
\\
m_k = \frac{1}{k} \sum_{j=0}^{k-1} m_j d'_{k-j}
\\
d'_j =
- \frac{j \beta}{2 \mu_0} \sum_{i=1}^2
\lambda_i v_i (\beta - v_i)^{j-1}
\left( \frac{\mu_0}{\beta \mu_0 + v_i (1 - \mu_0)} \right)^{j+1}
+ \left( \frac{\mu_0}{\mu_0 - 1} \right)^j
+ \sum_{i=1}^2 \frac{\nu_i}{2} \left( \frac{\mu_0 (\beta - v_i)}{\beta \mu_0 + v_i (1 - \mu_0)} \right)^j
$$
They have truncation error bounds here too (3.12).
Note: I'm not entirely sure why I typed all that out now, but, oh well; I did some simplifications. In the paper's notation of section 3 we have $n = 2$, $\alpha_i = v_i$, $\nu_i = 1$, $\nu = 2$, $p = 1$, $\delta_i = \lambda_i$.
