Sorry, you must first go through QC11.52 first.

Many people either stand up an additional ''jump box'' environment to upgrade to the intermediate version (11.52), then upgrade that to the newest version (12.20), or install 11.52 on the new (target) server and then do a further upgrade ''in-place'' on that server to 12.20.

Since QC10 is so old (and out of support by HP), it's DB version and OS version will likely not work with the newer 11.52 and certainly not with 12.20. Also, it is BEST PRACTICE to leave the old QC instance able to work (but with de-activated projects) during the upgrade in case anything goes wrong.
Check the System Requirements before you setup the NEW server and DB instance:
https://hpln.hp.com/page/alm-qc-enterprise-technical-specifications

You may want to setup this environment to work for QC11.52 OR 12.20.
I Suggest:

Microsoft Windows Server 2008 R2 Enterprise SP1 64 Bit

with

Microsoft SQL-Server 2008 R2 SP2

With that environment, you could optionally do an ''in-place'' upgrade to 12.20 on this same box, if you wanted.

**CAUTION:**

When you upgrade your projects from QC10 to 11.52, it also (automatically) does something after completing the UPGRADE step called REPOSITORY MIGRATION, which could take from a few minutes to a few days, depending on how large and how many artifacts are in your repository. The projects CANNOT be upgraded any further (i.e. into 12.20) until the repository migration is finished (shows 100% migrated in new Site Admin ''Repository Migration Status'' dialog from Tools menu).

Workaround: In order to save time ,suspend repository conversion in 11.52 at site admin level by setting the parameter ''suspend smart repository conversion'' to status of ''Y'',@ P T Burnam, I agree that we cannot upgrade a project from 11.52 to 12, when smart repository conversion is in progress for this project in 11.52,but there is a flexibility to suspend smart repository conversion in 11.52. set the parameter ''suspend repository conversion'' to Y as a workaround