The 24-year-old, who joined Forest for £1m from Iraklis Thessaloniki, smashed in Forest's opener in their 2-2 draw at Aston Villa on Sunday.
Vellios told BBC Radio Nottingham: "Yes, it was a special moment for me to score my first goal.
"I am very happy also as the goal helped us take an important point from an away game."
Vellios picked up the ball with his back to goal before turning to fire in from 22 yards and the former Everton forward had a special mention for the travelling 3,000 Forest supporters at Villa Park.
"The fans were brilliant," he added. "All of them helped us and this was an important point for us."
Forest return to action on Wednesday when they travel to Rotherham in the Championship and manager Philippe Montanier revealed striker Britt Assombalonga, who has been out of action since limping off against Wigan on 20 August, will miss out again.
"I don't think Britt will be fit," said Montanier. "I am not sure how long Britt will be out, perhaps it will be very quick but at this moment it is a little early for him."
Nottingham Forest striker Apostolos Vellios says his first goal for the club was a proud moment.