  White dwarfs (WDs) in active galactic nucleus (AGNs) discs might migrate to
the inner radii of the discs and form restricted three-body systems with two
WDs moving around the central supermassive black hole (SMBH) in close orbits.
These systems could be dynamical unstable, which can lead to very close
encounters or direct collisions. In this work, we use N-body simulations to
study the evolution of such systems with the different initial orbital
separation $p$, relative orbital inclination $\Delta{i}$ and SMBH mass $M$. It
is found that the close encounters of WDs mainly occur at $1.1R_{\rm H}
\lesssim p \lesssim 2\sqrt{3}R_{\rm H}$, where $R_{\rm H}$ is the mutual Hill
radius. For $p<1.1R_{\rm H}$, the majority of WDs move in horseshoe or tadpole
orbits, and only few of them with small initial orbital phase difference
undergo close encounters. For $p=3.0R_{\rm H}$, WD-WD collisions occur in most
of the samples within a time of $10^5P_1$, and considerable collisions occur
within a time of $t<62P_1$ for small orbital radii, where $P_1$ is the orbital
period. The peak of the closest separation distribution increase and the WD-WD
collision fraction decreases with an increase of the relative inclination. The
closest separation distribution is similar in cases with the different SMBH
mass, but the WD-WD collision fraction decreases as the mass of SMBHs
increases. According to our estimation, the event rate of the cosmic WD-WD
collision in AGN discs is about $300{\rm Gpc^{-3}yr^{-1}}$, roughly $1\%$ of
the one of the observed type Ia supernova. The corresponding electromagnetic
emission signals can be observed by large surveys of AGNs.
