Not without some prior preparation. There are proxy Xservers, such as XMX or xmove that would allow you to reconnect the proxy to a new Xserver in theory, thus when the displaying Xserver crashes, you can migrate to a new one, but they do have their limitations. XMX is designed for screencasting of a sort, and xmove is designed for user-initiated migration, not recovering from a crashed Xserver.
There are also applications like Xpra, and Xvnc that would allow you to run a second Xserver that does not display to the screen, then "attach" to those Xservers from another Xserver, thus if your displaying Xserver crashes, you can simply reattach.
Finally, there was a utility Guievict that used runtime code patching to migrate an application without its consent to another Xserver, but I find it doubtful that that would work in a modern environment.