Formatting a disk is not always equivalent to performing a manual TRIM on a drive - it depends if the format utility has explicit support for it. The Windows 7 format command does happen to have support for this. As for the difference between quick/full formats on Windows, if you do a quick format, it simply deletes the partition table and file listing. If you do a full format, it also does a full bad sector check.
Does it replace old data with new data or tells it to erase data just
like TRIM command without filling anything instead?
AFAIK, nothing is ever actually overwritten. I think the sectors are simply marked as free (unless you do a full format, which as overwrites each sector). Again, this is unless you're using an operating system with explicit TRIM support. From the blog post I linked to above:
Windows 7 requests the Trim operation for more than just file delete
operations. The Trim operation is fully integrated with partition- and
volume-level commands like Format and Delete, with file system
commands relating to truncate and compression, and with the System
Restore (aka Volume Snapshot) feature.
I expect that new builds of most Linux/Unix tools also have this support built-in, although you will need to verify this on a per-distribution basis (depending on TRIM support in the first place).
I want to use SSD on my system that seems to not allow TRIM so I would
just back up whole system, format and restore periodically to keep
spare blocks on SSD to help longevity and speed.
This usually will not suffice, as TRIM commands are far different from standard disk writes/reads/erase commands. If your operating system does not support automatically sending TRIM commands, you need to get a tool from your SSD manufacturer to manually TRIM the drive, or use another application like hdparam.
Final thoughts: If your operating system does not support TRIM, it would be wise to consider switching. While manually performing TRIM commands on the drive can increase the lifespan and performance of the drive (as opposed to simply doing nothing), using an operating system with explicit TRIM support will make these issues non-existent (and will make the drive last much longer than manually using a TRIM utility).