Flying blind here, no Qt environment to test with.

A quick scan of the Qt source: The qt_calendar_prevmonth and qt_calendar_nextmonth buttons are of class QPrevNextCalButton, which is a QToolButton. The painting code is only very slightly different to the underlying QToolBar, so it should be quite styleable. The icons used are QStyle::SP_ArrowRight and QStyle::SP_ArrowLeft. These seem to be set explicitly and may overwrite styled versions.

From the styling reference:
When the QToolButton displays arrows, the ::up-arrow, ::down-arrow, ::left-arrow and ::right-arrow subcontrols are used.
I think you also want the "image" (<Qt 5.15) or "icon" property (>= Qt 5.15) and probably the url() function.

I'd try something like one of these:
Qt Code: 
  1. QCalendarWidget QToolButton#qt_calendar_nextmonth::right-arrow {icon: url(right_arrow.png);}
  2. QCalendarWidget QToolButton::right-arrow {icon: url(right_arrow.png);}
  3.  
  4. QCalendarWidget QToolButton#qt_calendar_nextmonth {icon: url(right_arrow.png);}
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or equivakents using the "image" property.