= Solution
Let $s_j=\sin\beta_j$, $c_j=\cos\beta_j$. Inverting the two orthogonal <derivative> systems gives, on the vertical axis,
$$
q_x(0,y)=-s_1h_1(y)-c_1j_1(y),\qquad
q_y(0,y)=c_1h_1(y)-s_1j_1(y),
$$
and, on the horizontal axis,
$$
q_x(x,0)=c_2h_2(x)+s_2j_2(x),\qquad
q_y(x,0)=-s_2h_2(x)+c_2j_2(x).
$$
Using the <Wirtinger derivative> $q_z=(q_x-iq_y)/2$, these become
$$
q_z(iy)=-\frac{e^{-i\beta_1}}2[j_1(y)+ih_1(y)],\qquad
q_z(x)=\frac{e^{i\beta_2}}2[h_2(x)-ij_2(x)].
$$
Define the appropriate boundary transforms by
$$
\begin{aligned}
H_1(k)&=\int_0^\infty e^{ky}h_1(y)\,dy,&J_1(k)&=\int_0^\infty e^{ky}j_1(y)\,dy,\\
H_2(k)&=\int_0^\infty e^{-ikx}h_2(x)\,dx,&J_2(k)&=\int_0^\infty e^{-ikx}j_2(x)\,dx.
\end{aligned}
$$
The first pair is <holomorphic> for $\operatorname{Re}k<0$; the second pair consists of <Half-range Fourier transforms>, <holomorphic> for $\operatorname{Im}k<0$. Parametrizing the vertical boundary by $z=iy$ gives $dz=i\,dy$, which must be retained. Substitution yields the <quarter-plane oblique boundary spectral functions>:
$$
\boxed{\widehat q_1(k)=\frac{e^{-i\beta_1}}2[H_1(k)-iJ_1(k)],\qquad
\widehat q_2(k)=-\frac{e^{i\beta_2}}2[H_2(k)-iJ_2(k)].}
$$
Their signs follow from the stated clockwise boundary orientations: upward on the left and inward on the bottom.
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