Short answer
No. Jumuah replaces Dhuhr, so whoever prays Jumuah does not pray Dhuhr that day. This includes women and travellers who attend. The four and two rakats after Jumuah are its sunnah, not Dhuhr. If you missed Jumuah, pray four rakats of Dhuhr instead.
Jumuah takes the place of Dhuhr on Friday for everyone who prays it, including those not obliged to attend. A "precautionary Dhuhr" (ihtiyat al-zuhr) after a valid Jumuah is not required, and Hanafi muftis generally tell people not to pray it in towns and cities where Jumuah is properly established.
The four schools agree on this.
Explanation
Jumuah is Friday's prayer in place of Dhuhr. It is prayed in Dhuhr's time, as two rakats after a khutbah, and it fulfils the duty of the midday prayer. The four schools agree that someone who prays Jumuah does not pray Dhuhr as well.
- Women and travellers are not required to attend, but if they pray Jumuah with the imam, it counts and they do not pray Dhuhr.
- The rakats after Jumuah (four, then two in the Hanafi school) are sunnah, not Dhuhr; see how many rakats are in Jummah.
- Missed Jumuah: pray four rakats of Dhuhr; see what happens if you miss Jummah.
- At home alone: pray Dhuhr; see can you pray Jummah at home.
You may have seen people pray four extra rakats called "ihtiyat al-zuhr" (precautionary Dhuhr) after Jumuah. Some Hanafi scholars of the past suggested it in places where they doubted that the conditions of Jumuah were met, such as small villages. In towns and cities where Jumuah is properly held, Hanafi muftis say not to pray it, so that people do not start to think Jumuah is not enough. If you live somewhere small and are unsure, ask a local scholar.
The full method is in how to pray Jummah.
The evidence
Quran, Surah Al-Jumu'a 62:9
يَٰٓأَيُّهَا ٱلَّذِينَ ءَامَنُوٓا۟ إِذَا نُودِىَ لِلصَّلَوٰةِ مِن يَوْمِ ٱلْجُمُعَةِ فَٱسْعَوْا۟ إِلَىٰ ذِكْرِ ٱللَّهِ وَذَرُوا۟ ٱلْبَيْعَ ۚ ذَٰلِكُمْ خَيْرٌۭ لَّكُمْ إِن كُنتُمْ تَعْلَمُونَ ﴿٩﴾
O you who have believed, when [the adhan] is called for the prayer on the day of Jumu'ah [Friday], then proceed to the remembrance of Allah and leave trade. That is better for you, if you only knew.
Jumuah is the prayer called for on Friday at midday.
Sahih al-Bukhari 904
Sahih... كان يصلي الجمعة حين تميل الشمس ...
"He used to pray Jumuah when the sun declined from its zenith."
Jumuah is prayed in the time of Dhuhr, in its place.
Where the ruling is recorded
- al-Hidaya, Kitab al-Salah, chapter on Jumuah. One who is excused but prays Jumuah has it count in place of Dhuhr.
- Radd al-Muhtar (Ibn Abidin), Kitab al-Salah, chapter on Jumuah. Discusses the four rakats some prayed as a precautionary Dhuhr, and that ordinary people should not be told to pray them.
This answer is pending review by the scholars on our team. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.