Upon carefully reading through terrain bond, which I will post for everyone's convenience;
At 4th level, the guide forms a bond with the land itself, enabling him to direct others in such terrain. When in his favored terrain, the ranger grants all allies within line of sight and that can hear him a +2 bonus on initiative checks and Perception, Stealth, and Survival skill checks. Also, as long as they travel with him, the ranger's allies leave no trail and can't be tracked. The ranger can choose for the group to leave a trail, or even specific members of the group to leave a trail if he so desires. This ability replaces hunter's bond.
Tentatively, I would agree with your player on what the ability technically states. The idea that you and your allies pass without trace is separate from the bonus to initiative, Perception, Stealth, and Survival which are in a sentence that specifically states that you must be in your favored terrain. The sentence that grants pass without trace is added after a period and does not again reference being in that terrain. Honestly the ability could be slightly better worded.
However, as all of the above are abilities granted by favored terrain, and terrain bond's first sentence encourages the thought that you are sharing these bonuses, and the sentence about granting pass without trace is listed after the qualifying sentence about being in your favored terrain I would confident in stating that the intention of the ability is only to function in your favored terrain.
Since you seem to be a GM by the phrasing of "a player of mine," I would encourage disallowing the use of terrain bond outside of favored terrain.