Newquay parish was created in 1882, prior to this Newquay was a hamlet in St. Columb Minor, so the parish records of St. Columb Minor would be where you should look for records of him, although it is possible he was recorded in some neighbouring parish.
If the family were non-conformists, however (Wesleyan or other), it may be more complicated to figure out if/how/where his birth was recorded.
http://www.cornwall-opc-database.org/search-database/baptisms-non-conformist/While I can't see any direct evidence of your Henry here, a Henry and Mary Thomas had a son Evan John baptised in 1840, and a son Mark baptised in 1842, in the Columb, St. Wesleyan Methodist Circuit (would likely cover Newquay)
According to the GRO index, the mother's maiden name was Bennick or Binnock.
Having trouble finding them on the cenuses. Mark died in Canada, looks like:
https://www.familysearch.org/ark:/61903/1:1:JKW6-6LGInterestingly, says he was 84 years and 8 months old and had been in Ontario for 80 years - so were the whole family in Canada by 1851?
Also that his parents were from Wales.
Another possible brother, Phillip b. 1834, married in Canada in the 1870s:
https://www.familysearch.org/ark:/61903/1:1:FMN4-SX1He was born in Devon according to this.
If John here is "Evan John" this might be the family, but no Henry?
https://www.familysearch.org/ark:/61903/1:1:MQB3-NXVOn Freecen it appears the older children were born out of county (Devon?) as were the parents
