The Taylor series for two variables up to quadratic terms is:$ f(x, y) = f(x_0, y_0) + (x- x_0) \dfrac{\partial f}{\partial x} + (y - y_0) \dfrac{\partial f}{\partial y} + \dfrac{1}{2} \left [ (x - x_0)^2 \dfrac{\partial^2 f}{\partial x^2} + 2 (x - x_0)(y - y_0) \dfrac{\partial^2 f}{\partial x \partial y} + (y - y_0)^2 \dfrac{\partial^2 f}{\partial y^2} \right ] + \cdots$ The calculations are tedious if done by hand. For example$ \frac{\partial f}{\partial x}=\frac{\sinh (x) \cosh (y)}{\sqrt{2 (x-y)+1}-0.5}-\frac{\cosh(x) \cosh (y)}{\left(\sqrt{2 (x-y)+1}-0.5\right)^2 \sqrt{2(x-y)+1}}$which evaluated at (0,0) is -4.
Using software I found that:$ f(x,y)=2-4x+4y+11x^2-20xy+11y^2+\cdots$
EDIT
Following the comments here is a more detailed calculation of the expansion of $f(x,y)=\sqrt{2(x-y)+1}$. The derivatives are easy:$ f_x=-f_y=\frac{1}{\sqrt{2 (x-y)+1}} ; f_{xx}=f_{yy}=-f_{xy}=-\frac{1}{(2 (x-y)+1)^{3/2}}$
Now evaluate those at (0,0) and you get the expansion:$f(x,y)=1+x-y-x^2/2-y^2/2+xy+\cdots$