The Supported Upgrade Paths document outlines the min. number of firmware upgrades you will need to follow to upgrade the 60C -- ideally, you will want to get the 60C to the same firmware to match the 60D.
Glancing at the quick start guides for both the 60C and 60D it looks like they have the same ports, so you should be able to save an unencrypted backup config from the 60C, replace the top line of the config with one from the 60D, load that into the 60D. After you load the config onto the 60D, perform a "diag debug config-error-log read" on the CLI to check for any errors.
Fortigate 60C and 60D are very similar, 60C has only 5 internal lan ports, 60D has 7
Importing the config should still work, assuming the LAN ports are similarly labelled in both devices.
Things to watch out for when you upgrade between major firmware releases (e.g. going from 4.0 MR2->4.0 MR3->5.0, etc) is certain features may have be bumped/moved around/or removed. Read the patch notes (e.g. 4.0 MR3 patch1, 5.0.1, 5.2.1, etc.) to see what was removed/need to fix.
Do perform "diag debug config-error-log read" on the CLI after each upgrade.
Also check/confirm you have no spaces or other non-standard chars in your firewall address/labels/names, etc. as the "scripted conversion process" performed on the config during the firmware upgrade can mess up on those. e.g. a web filter name like "This is my web filter" could get truncated down to "This".
Personally, with 5.x being so different to previous firmware releases, I would just rebuild the config from scratch, using the old config as a template. Since you have access to both devices, you can open the GUIs side-by-side each to other while you code/build the new config on the 60D. Rebuilding the config from scratch will also remove any obsolete or unused settings still left on the old config and also gives you the chance to streamline it or add something that you couldn't do before. YMMV.