Picard opens the manual non-sliding glass door for him.
Here's Wil's reference:
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The episode is "Samaritan Snare" from the second season. The door was the hinged all-glass type you might find inside a present-day office building. Picard and Wesley both approached the door and both stopped as if expecting the other to open it. Finally Picard opened the door and Wesley held it open for him. The door had a large metal handle so it seems unlikely that either of them were confused about how to open it.