No; sharing damage is not one of the properties of the Elder Brain's Psychic Link.
The Elder Brain's Psychic Link trait (VGtM, p. 174) says:
Psychic Link. The elder brain targets one incapacitated creature it can perceive with its Creature Sense trait and establishes a psychic link with that creature. Until the psychic link ends, the elder brain can perceive everything the target senses. The target becomes aware that something is linked to its mind once it is no longer incapacitated, and the elder brain can terminate the link at any time (no action required). The target can use an action on its turn to attempt to break the psychic link, doing so with a successful DC 18 Charisma saving throw. On a successful save, the target takes 10 (3d6) psychic damage. The psychic link also ends if the target and the elder brain are more than 5 miles apart, with no consequences to the target. The elder brain can form psychic links with up to ten creatures at a time.
For reference, its Creature Sense trait makes it aware of any intelligent creatures within 5 miles of it.
It also has a Sense Thoughts action that lets it read the thoughts and feelings of the psychic-linked creature. It also has several legendary actions affecting a psychic-linked creature that allow it to break the creature's concentration automatically, do psychic damage to all creatures near the target, or sever the psychic link (which gives the target disadvantage on all ability checks, attack rolls, and saving throws until the end of the linked creature's next turn).
None of these indicate that the Elder Brain shares in the damage it receives. In particular, the fact that it can do psychic damage to enemies of the elder brain near that creature as a legendary action would make such damage-sharing actively harmful to itself, so it wouldn't make much sense for it to share in that damage.