'tried alot' means what? If you meant, kept repeating the sending via Google and it continued to fail, then there was nothing changed ... thus nothing really different.
That aside ... using Google for a relay. Even if you have correct credentials, Google might still not allow relay because of the DNS and spam checks it makes.
Using Google, might require some additional records to the your domain name of the moodle server. An SPF record in the DNS for your domain that includes the IP address of the moodle server might be all that is needed.
Since you are using Google:
https://support.google.com/a/answer/33786?hl=en
Here's a tool that will help generate an SPF record.
What it generates one needs to add to the zone for vumoodle.org
The DNS servers that are authorative for that zone are:
vumoodle.org. 86400 IN NS ns5.ctrls.in.
vumoodle.org. 86400 IN NS ns4.ctrls.in.
So it's in those DNS servers the SPF records need be added for your zone.
One way to see how messages look sent to google ... send a message that that google account.
When viewing the message use the down arrow icon and select 'Show Original'
You'll see all the mail servers the message ran through along with references to DKIM, SPF, etc.
'spirit of sharing', Ken