The pair were sent to the stand during County's 2-1 home Scottish Premiership win on February 22.
Their bans have been suspended until December 21 but both were warned by the Scottish Football Association's Disciplinary Tribunal that they will kick in if there is any repeat of the dug-out spat.
Speaking before his Hampden hearing on Thursday afternoon, Lennon said: "I will go and take my punishment on the chin and that will be it hopefully."
Lennon reacted after Adams went on to the pitch to complain about Saints skipper Jim Goodwin's aerial challenge on Melvin de Leeuw, when the defender caught the back of the County winger's head with his forehead.
Goodwin had earlier enraged Adams when he caught Fillip Kiss with an arm in another aerial challenge.
Adams was accused of adopting a "threatening and aggressive attitude towards an opposing member of team staff", while Lennon has received a notice of complaint alleging the same offence along with "the repeated use of offensive, insulting and abusive language".
After being found guilty of the offence, both men were given two-match bans, suspended until December 31 of this year, which will kick-in if they breach Disciplinary Rule 203 - which prohibits managers engaging in misconduct on a matchday - during that period.
The tribunal has also warned both bosses to "be mindful of remaining in their own technical areas in future".