You can't get the millis directly from a Period, since fields like months and years are variable in terms of milliseconds.
In order to make this work, you need to supply a "baseline" instant from which Period can calculate that actual millisecond duration.
For example, the Period.toDurationFrom and Period.toDurationTo methods take such a baseline instant, and calculate a Duration object, which you can then obtain the millis.
The Javadoc for toDurationFrom says:
Gets the total millisecond duration of this period relative to a start instant.
  This method adds the period to the specified instant in order to calculate the duration.
An instant must be supplied as the duration of a period varies. For example, a period of 1 month could vary between the equivalent of 28 and 31 days in milliseconds due to different length months. Similarly, a day can vary at Daylight Savings cutover, typically between 23 and 25 hours.
So you need to pick an appropriate baseline instant for your application.