Date is exactly the example given in https://developer.android.com/training/data-storage/room/referencing-data. 
For example, if we want to persist instances of Date, we can write the following TypeConverter to store the equivalent Unix timestamp in the database:
public class Converters {
    @TypeConverter
    public static Date fromTimestamp(Long value) {
        return value == null ? null : new Date(value);
    }
    @TypeConverter
    public static Long dateToTimestamp(Date date) {
        return date == null ? null : date.getTime();
    }
}
The preceding example defines 2 functions, one that converts a Date object to a Long object and another that performs the inverse conversion, from Long to Date. Since Room already knows how to persist Long objects, it can use this converter to persist values of type Date.
Next, you add the @TypeConverters annotation to the AppDatabase class so that Room can use the converter that you've defined for each entity and DAO in that AppDatabase:
AppDatabase.java
@Database(entities = {User.class}, version = 1)
@TypeConverters({Converters.class})
public abstract class AppDatabase extends RoomDatabase {
    public abstract UserDao userDao();
}
A side note: java.util.Date is considered to be badly designed (and java.util.Calendar is much worse). If you have any non-trivial date-time logic and can get away with API level 26 (Java 8 on desktop), it's generally better to use java.time package. And if you can't, see https://github.com/JakeWharton/ThreeTenABP for a backport.