No, the English word "York" comes from Old Norse Jórvík, named so by it's Viking settlers. The Welsh word "Efrog" comes from the Brythonic word Eborakon and the Latin name for the city was also just a Latinisation of Eborakon, Eboracum called so by the Romans.
The name Jórvík became York over time after the Norman conquest.
The Brythonic/Welsh translation is "Place of Yew Trees".
Actually, PauBofill is right. The Proto-Celtic Eborakon (meaning "yew-place") was Latinized by the Romans as Eboracum. The Anglo-Saxons reinterpreted Eboracum as Eoferwic (meaning "boar-settlement"). The Danish conquerors of Eoferwic (which would have sounded a bit like "yo(v)erwick") respelled this as Jórvik, which in time got worn down to just "Jórk" -- or, in later English spelling, York. So there is a straight line of descent (even if there have been one or two misunderstandings along the way!), with today's Welsh Efrog being simply the modern form of the Proto-Celtic Eborakon.
You're not mistaken. The Danish/Norse "Jorvik" comes from Eboracum by a process by which the word ending was reduced and -vik (=town, settlement) added. No one really knows what "Eboracum" meant but in Gaul -acum (Latin) was often added to known names to mean "estate of".