One of the sisters told me that if a person did not pray Fajr on time, i.e., he prayed it after sunrise, he cannot offer the Sunnah prayer of Duha, i.e., no one can offer the Sunnah prayer of Duha who did not offer (Fajr) prayer on time. Is this correct?

Praise
be to Allah.

Firstly: 

Duha
prayer is a confirmed Sunnah (Sunnah mu’akkadah); it is proven that
the Prophet (sa) did it, as Muslim (1176) narrated from ‘Aa’ishah (may Allah
be pleased with her) that she said: The Messenger of Allah (blessings and
peace of Allah be upon him) used to pray Duha with four rak‘ahs, and more as
Allah willed. 

An-Nawawi
(may Allah have mercy on him) said: Our companions said: Duha prayer is a
confirmed Sunnah (sunnah mu’akkadah).

End
quote from al-Majmoo‘, 3/528 

Shaykh Ibn Baaz (may Allah have mercy on him) said in Majmoo‘ al-Fataawa
(11/389): Duha prayer is a confirmed Sunnah (sunnah mu’akkadah) that the
Prophet (sa) did and taught his companions to do. End quote. 

For
more information on the hadeeths that speak of its virtue, please see the
answer to question no. 145070 

Secondly: 

If a
person missed Fajr prayer, but made it up after sunrise, then he may pray
Duha, because Duha prayer is a confirmed Sunnah (sunnah mu’akkadah) that has
no connection to Fajr prayer, and has nothing to do with the time of Fajr
prayer at all, or whether it was done on time or made up later. Rather what
is prescribed for one who has fallen short in some deed, or missed out on
something good, is to do a lot of good and righteous deeds, the best of
which are the naafil (supererogatory) prayers. 

It
was narrated that Rabee‘ah ibn Ka‘b al-Aslami (may Allah be pleased with
him) said: I used to stay overnight with the Messenger of Allah (blessings
and peace of Allah be upon him) and bring him water for wudoo’ and whatever
he needed. He said to me: “Ask.” I said: I ask to be with you in Paradise.
He said: “Is there anything else?” I said: That is all, He said: “Help me to
do that for you by prostrating a great deal.”

Narrated by Muslim in his Saheeh, 489 

An-Nawawi
(may Allah have mercy on him) said: This indicates that it is encouraged to
prostrate a great deal; what is meant is prostration in prayer.

End
quote from Sharh Muslim. 

It
was narrated that Ma‘daan ibn Abi Talhah said: I met Thawbaan, the freed
slave of the Messenger of Allah (blessings and peace of Allah be upon him),
and said: Tell me of a deed that I may do, by which Allah may admit me to
Paradise – or he said: Tell me of the most beloved of deeds to Allah. He
remained silent. I asked him (again) and he remained silent. I asked him a
third time and he said: I asked the Messenger of Allah (blessings and peace
of Allah be upon him) about that and he said: “You should prostrate to Allah
a great deal, for you will not perform one prostration to Allah but Allah
will raise you one degree in status thereby and erase one sin for you.”

Narrated by Muslim in his Saheeh, 488 

And
Allah knows best.