After I compared your sample with my code I solved the issue by add mode two-way to the "IsOpen"
old code:
<syncfusion:SfPicker x:Name="MonthPicker" ItemsSource="{Binding ListMonths}" DisplayMemberPath="MonthString" IsEnabled="True"
IsOpen="{Binding IsOpen}"
HorizontalOptions="Center"
PickerHeight="300"
PickerMode="Dialog"
PickerWidth="280"
HeaderText="Select a month"
ShowHeader="True"
ShowFooter="True"
HeaderBackgroundColor="#2499ce"
ColumnHeaderTextColor="White"
SelectedItemTextColor="#2499ce"
UnSelectedItemTextColor="Black"
VerticalOptions="Center">
</syncfusion:SfPicker>
replaced with :
<syncfusion:SfPicker x:Name="MonthPicker" ItemsSource="{Binding ListMonths}" DisplayMemberPath="MonthString" IsEnabled="True"
IsOpen="{Binding IsOpen,Mode=TwoWay}"
HorizontalOptions="Center"
PickerHeight="300"
PickerMode="Dialog"
PickerWidth="280"
HeaderText="Select a month"
ShowHeader="True"
ShowFooter="True"
HeaderBackgroundColor="#2499ce"
ColumnHeaderTextColor="White"
SelectedItemTextColor="#2499ce"
UnSelectedItemTextColor="Black"
VerticalOptions="Center">
</syncfusion:SfPicker>
thanks