My desktop computer motherboard is not working due to power issue. I would like to change the motherboard. Please note that my processor is core i7-620M. I would be very pleased is anyone suggest me a motherboard which is compatible with the processor.
Well, your processor is already eight years old:
https://ark.intel.com/products/43560/Intel-Core-i7-620M-Processor-4M-Cache-2_66-GHz Intel® Core™ i7-620M Processor (4M Cache, 2.66 GHz) Product Specifications
And, without more information, your processor may be soldered to the motherboard.
You may be looking at a motherboard failure, or power supply failure.
Intel no longer makes desktop boards. So, you will have to look at other manufacturers and the used market, if your processor is socketed, to find a board, and that is only if the bios of the replacement board is compatible.
It makes little sense to go this path, with such an old system.
It is best to get a new desktop, which means replacing all of the hardware. If the board is bad, the memory, power supply, processor, AND DISKS cannot be too far behind.
Backup you data now and get a new system.