Through a bit of trial and error I found the solution using Chrome Developer console. For anyone else having this problem here are the CSS settings that I used. You’ll probably have to change the font to one that you have installed, the Arsenal font that I am using required a Google Fonts plugin to get it working.
/* Calendar title */
#tribe-geo-results h2.tribe-events-page-title, h2.tribe-events-page-title {
font-family: Arsenal;
}
/* Calendar days */
.tribe-events-calendar thead th {
font-family: Arsenal;
}
/* Calendar dates */
.tribe-events-calendar td div[id*=tribe-events-daynum-] {
font-family: Arsenal;
}
/* Calendar popup description */
.tribe-events-calendar p {
font-family: Arsenal;
}
/* Calendar popup title */
.tribe-events-calendar h4.entry-title.summary {
font-family: Arsenal;
}
/* Popup calendar start end time */
.tribe-events-calendar abbr.tribe-events-abbr.tribe-event-date-start {
font-family: Arsenal;
}