There were few clear-cut chances in the rearranged fixture at the Stade Louis II, but Lucas had the ideal opportunity to break the deadlock in the 52nd minute when Nabil Dirar felled Bryan Dabo in the area.
However, the forward fluffed his lines from 12 yards with a wild effort. A draw means Monaco are now two points adrift of the UEFA Champions League places, while Montpellier remain seventh.
Head coach Leonardo Jardim lamented Monaco's slow start in last Friday's 1-1 draw against Saint-Etienne and his side failed to find their rhythm in a cagey opening.
Indeed it was Montpellier who started to show more going forward. Dabo's dipping free-kick went just wide of the right-hand post, before Lucas was unable to sort his feet out to tap home the rebound after Danijel Subasic had saved Morgan Sanson's initial effort.
Monaco improved towards the end of the half and it appeared as though they had made the breakthrough when Layvin Kurzawa nodded home a free-kick from the left, only for his celebrations to be rightfully curtailed by the offside flag.
The visitors had the ideal chance to move in front seven minutes after the break when Dirar was adjudged to have tugged Dabo to ground in the penalty area, but Lucas sent his penalty ballooning high and wide of the right-hand post.
As the clock wound down, Monaco saw tentative penalty claims for handball against Abdelhamid El Kaoutari waved away, while Bernardo Silva wasted a great chance when he shot straight at Geoffrey Jourdren from near the penalty spot.
