No the Hail Holy Queen is not part of the rosary. The rosary ends with the last Glory be. (technically the Fatima prayer is not part of the rosary as well).
To be extremely technical - and this often surprises people - the intial Creed, Our Father, 3 Hail Mary’s and Glory be are not part of the rosary either - though since now the practice is so well entrenched, everyone regards it as so.
The reason most people say the Hail Holy Queen, was because the rosary was often regarded as a replacement for a part of the Liturgy of the Hours ( or ‘Office’). Because at that time, the Hail Holy Queen was said through most of the year after every choral ‘office’, it was said after the rosary as well. At one point it also common to begin the rosary in the same way as the Liturgy of the Hours(i.e. O God, come to my aid…) . This was how it was begun:
Hail Mary full of grace the Lord is with you.
- Blessed are you among women.
Lord open my lips
And my mouth shall declare your praise
God, come to my aid
Lord make haste to help me.
Glory be to the Father, etc. Alleluia
or in Lent:
Glory be to the Father, etc. Praise to you Lord, King of eternal glory.