Not without writing a program that sits and watches both databases and compares them at regular intervals.
2-way replication is very tricky because you have to deal with issues like what if the same record is changed in both databases; who wins?
I'm not familiar with Filemaker but if it has it's own language and can read and write ODBC sources, your best bet may be to write a program in Filemaker to watch itself and update MySQL when necessary.
HTH,
=C=