The dates of the solstices and equinoxes can be determined by observing the apparent north-south position of sunrise from a fixed observation point over the course of a year.
The easiest way to do this, if you are left in peace to make your observations, is to erect a vertical pole where it casts a shadow at sunrise onto an east-facing wall. Mark the position of the shadow at each sunrise, and carefully count the days.
Over the course of several months, you will observe the apparent position of sunrise to march north (or south) day by day. This march will slow and eventually reverse; the day of reversal is the day of summer (or winter) solstice.
Actual solstice is almost impossible to observe directly and in real time, because so many consecutive sunrises seem to occur in the same place. However, after the solstice, the position of sunrise will migrate back over your earlier observations. Choose some earlier sunrise position and note the number of days that elapse before the sunrise returns to it. The solstice was exactly halfway through that period.
The equinox will be 91 days after the solstice.
For complete accuracy, you must repeat these observations over the course of many years. If you are in contact with settled or semi-nomadic locals, they may have already made these observations and are maintaining a calendar, so they can tell you the date of the next solstice or equinox.
Look for a society who have erected a vertical pole in an open space east of a north-south wall.