In Dumas's The Count of Monte Cristo, Albert de Morcerf is son to Fernand Mondego—the Count de Morcerf—and Mercédès, who was Edmond Dantès' former sweetheart.

In Chapter 37, Albert has been taken prisoner and is being held for ransom by the Italian bandit, Vampa. Vampa (a friend of Monte-Cristo) is demanding money for the safe release of Albert. Albert sends a letter to Franz with instructions on how to free him. In the letter, Albert signs his full name:

My Dear Fellow,—The moment you have received this, have the kindness to take the letter of credit from my pocket-book, which you will find in the square drawer of the secretary; add your own to it, if it be not sufficient. . . It is urgent that I should have this money without delay. I do not say more, relying on you as you may rely on me. Your friend,

Albert de Morcerf.

In Chapter 112, Mercédès speaks to Monte Cristo about his original attempt to harm “the son of M. de Morcerf,” who is also Mercédès’s son.

"Alas," said Monte Cristo, "your words sear and embitter my heart. . . I have been the cause of all your misfortunes; but why do you pity, instead of blaming me? You render me still more unhappy."

"Hate you, blame you—you, Edmond! Hate, reproach, the man that has spared my son's life! For was it not your fatal and sanguinary intention to destroy that son of whom M. de Morcerf was so proud? Oh, look at me closely, and discover if you can even the semblance of a reproach in me." The count looked up and fixed his eyes on Mercédès, who arose partly from her seat and extended both her hands towards him. 

By the end of the story, Albert learns the truth about his father's (Mondego) reprehensible actions that betrayed the pasha of Janina to his enemy. Count Mondego is publicly disgraced and ultimately takes his life. Albert and his mother donate the funds of the estate to charity, and Albert leaves to become a soldier. Mercédès is left alone, but the young Edmond Dantès had saved money for Mercédès prior to their separation. He gifts her the money so she can survive.

It is important to note that the movie version is very different from Dumas's story, so one needs to be careful not to draw an understanding of the novel from the film.

No. Albert de Morcerf is the son of the Count and Countess de Morcerf. See a listing and the relationships of the characters within the novel at the link: