I am using the Arria II GX transceiver block. Is it possible to change the transfer rate dynamically?

Is possible.

By using ALTGX_RECONFIG, you can change various functions of the transceiver block without reconfiguring the FPGA.

When changing the transfer rate, generate each transfer rate setting in MIF file format and save it as the initial value of the FPGA internal RAM. For example, RAM1 is 1.25 Gbps, RAM2 is 2.5 Gbps, RAM3 It is recommended to store an image such as 3.125 Gbps in FPGA internal RAM (actually it will function as ROM) and dynamically reconfigure it by switching it with user logic.

There is a function that automatically generates the information of the internal setting register of the transceiver block in the MIF file in the Quartus II option settings, so you can easily achieve this by using it.

For more information, please refer to the document at the URL link below.
   https://www.altera.com/en_US/pdfs/literature/an/an558.pdf