Yes, you can put FIFOs into your FPGA. You can design your own module by using ordinary FPGA RAM plus some control logic. Or you can use a pre-made module such as these cores from Xilinx (scroll down to Memories & Storage Elements):
**broken link removed**
FIFOs are very common, so Xilinx put FIFO control logic into the block RAMs of its new Virtex-4 FPGAs.