Is there a way to dump timing reports to a file?

Your .sdc file should only have constraints and not any timing analysis commands like report_timing. (Note that the .sdc files are used through out the compile, many places which don't make sense to use timing calls. Quartus used to error out on these. I don't think it does any more, but the recommendation is to separate them into a .tcl file).

You can:

a) Add a .tcl file to run as part of the compile. This can be added under Assignments -> Settings -> TimeQuest. Of course it's a .qsf setting, which is what you'll want to add, I just don't know if off hand.

b) If you're running quartus_sh to do your compile, you can always call quartus_sta separately, and have it run a special script.