Configure (upload bitstream) to MAX10 without Altera tools using Linux
I think the standard method for doing this is to generate an SVF or XSVF file with the Quartus software and then using something like OpenOCD or http://www.clifford.at/libxsvf/ to play it back. You would interface to the FPGA via the JTAG pins, either via GPIO or perhaps an FTDI chip. 

Another option would be to forego the FPGA's onboard configuration storage entirely and directly load the configuration during boot via SPI or similar, presuming the FPGA supports it. This way, there is no need to upgrade the FPGA firmware during the firmware upgrade process as it would be integrated with the main firmware and loaded into the FPGA automatically.