This depends on your point of view. The British colonial administrators in the late nineteenth and early twentieth centuries tended to use the word "civilized" to reflect their own value system. They tended to think that "civilized" meant white Christians dressing in European style and having a highly literate European style of bureaucracy. Of course, the Igbo were not British and did not fit that model. The Igbo recognized cultural differences and the diversity of different peoples, as seen when when Achebe states:
The world has no end, and what is good among one people is an abomination with others.
Even though the Igbo were not "civilized" from the narrow viewpoint of the British in the novel, the Igbo had their own civilization, with their own religion, oral traditions, modes of governance, and customs. The Igbo thus were what an objective reader would call civilized. Much of the tragedy of the book deals with the breakdown of Igbo civilization as a result of colonialism and how Igbo individuals are caught between the two cultures.
The question of civilization is central to the conflict between the Igbo and the British. It goes back to Kipling's "White man's burden"- the idea that Europeans and Westerners in general (and the British in particular) were responsible for "civilizing" and converting natives on other continents, who were in turn called savages. This is in part why Africa was known as "the dark continent." Because the societies were structured differently from those in Europe, they were not seen as civilized in their own right.
The Igbo villages demonstrate each aspect of what is considered civilization: systems of government, economy, and justice, social/communal roles and ceremonies, communication/diplomacy with other societies, etc. In addition, they have established religion, art, and elaborately developed oral storytelling. That is one purpose in Achebe's writing, to bring knowledge of this culture to a wider audience, and to combat the stereotype of pre-colonial Africa as "uncivilized."
The shortest answer is yes. To the question of whether or not the Igbo people are "civilized" in Things Fall Apart, we simply need to ask if the Igbo have social rules that govern behavior, distinguishing proper behavior from improper and providing a framework for social/population growth.
Since the concept of civility is basically defined in accordance with the presence of social rules and the Igbo's sometimes elaborate and often strictly enforced social rules play a central role in the text, we can say with certainty that the Igbo people are civilized.
For example, the head of a household honors any male guest by praying over and sharing a kola nut with him, offering the guest the privilege of breaking the nut. They dank palm-wine together, with the oldest person taking the first drink after the provider has tasted it (eNotes).
If the English or Japanese practices of taking tea/tea rituals are often seen as standard, symbolic examples of civilized behavior, we can draw a strong and direct connection from tea rituals to those of the kola nut.
Social norms like this one and many others help to define the traditions and social life of the Igbo culture. Other norms regulate marriage, trade, agriculture and conflict.
The conflict of colonialism that occurs in the late portions of the novel then becomes one of a clash of civilizations - not a conflict between one civilized people and one uncivilized people.