Without specific logic implemented in the MAX10, JTAG is the only interface available for flash programming. Quartus only supports flash programming through Altera programming adapter, but it's possible to use generic JTAG programming files with third party tools. Some may be able to implement a JTAG programmer through FTDI USB adapter.
The other option is use a dual flash configuration and program the application part of the flash in remote system upgrade mode. The data can be streamed to the MAX10 device via any interface of your choice, e.g. UART.
Both options are discussed in detail in the MAX10 handbook chapter "MAX 10 FPGA Configuration Design Guidelines".