and it claimed that African countries have an unspoken agreement to not alter national borders in fear of inciting ethnic conflict. Is this true?

Nope, it is not an unspoken agreement. Article III paragraph 3 of the OAU charter affirms member states commitment to the principle of

Respect for the sovereignty and territorial integrity of each State and for its inalienable right to independent existence.

While Article III paragraph 4 commits to

Peaceful settlement of disputes by negotiation, mediation, conciliation or arbitration

which opens the door to border adjustments arrived at through a negotiated settlement agreeable to all parties.

Article VI further underscores these points by stating

The Member States pledge themselves to observe scrupulously the principles enumerated in Article III of the present Charter.

At the 1964 Organization of African Unity meeting in Cairo, the body adopted a resolution concerning border disputes among member states. The resolution notes problems that border disputes could raise, such as the possibility of outside meddling by Non-African powers, or becoming a cause for dissention within the OAU.

For those reasons, the resolution finishes with the following statements:

1 [The OAU] SOLEMNLY REAFFIRMS the strict respect by all Member States of the Organization for the principles laid down in paragraph 3 of Article III of the Charter of the Organization of African Unity;

2 SOLEMNLY DECLARES that all Member States pledge themselves to respect the borders existing on their achievement of national independence.

So, neither the charter nor the 1964 resolution on borders explicitly mention the possibility of ethnic conflict or secessionist movements. Instead, they focus on the possibility of state vs state border conflict.

However, from an IR perspective, it is usually agreed that African heads of state, and the OAU/AU as a body have chosen not to support or recognize secessionist efforts because of the threat they pose to peace and stability on the continent. There are very few African states that are without ethnic minorities. So, the president of Nigeria, for instance, wants fixed borders because he doesn't want secessionist rebels in Nigeria to have a hope that they can break away and form a new state.

[Edit- one notable exception to this rule was the case of the Nigerian civil war, where Tanzania, Gabon, Cote d'ivoire and Zambia all extended diplomatic recognition to Biafra. However, the majority of OAU member states continued to recognize Nigeria as the sole legal representative, and at the OAU summit in 1967 resolution 51 was passed reaffirming a commitment to the territorial integrity, unity and peace of Nigeria.]

Of course, there have been new states formed and new borders drawn since 1964. Eritrea successfully seceded from Ethiopia in 1991, after 30 years of independence struggle. Of course, this happened in the specific circumstances where Eritrean and Ethiopian rebels overthrew the Derg regime. Thus, Eritrean independence could be presented as a mutual agreement between the Eritrean Liberation Front and the new government of Ethiopia in 1991. Further, the post 1991 borders of Eritrea followed the colonial era borders of Italian Eritrea. That, combined with the fact that Eritrea was federated with Ethiopia from 1952-1962 before annexation, allowed Eritrean leadership and African heads of state the ability to argue that Eritrea had existed as an independent state, and post 1991 Eritrea was a return to this historical independence that had been abrogated in 1962.

The case of South Sudan is unusual, in that there was not a historical state in the post-independence era that could be pointed to in order to justify South Sudanese separation. South Sudan independence represented the drawing a new "internal border" with Sudan. It also represented a break with a tradition of not recognizing secessionist efforts. However, the two Sudanese Civil Wars from 1955-1972 and 1983-2005 undoubtedly played a role in convincing AU heads of state that political reconciliation between the north and south was not likely after 39 years of civil war in a 50 year period. It was also important that the Comprehensive Peace Agreement did not mandate independence. Rather, it committed the government of Sudan and the SPLM to hold a referendum on independence. Thus, the decision of the referendum to seek South Sudanese independence could be justified by Article III paragraph 4 of the OAU/AU charter, as part of a negotiated settlement. To say it another way, by agreeing to hold the referendum, Sudan agreed in principal to recognize South Sudanese independence.

Those two exceptions aside, there have been a large number of secessionist movements that have received little to no international recognition, and the AU appears to view the case of South Sudan as a one-off, not constituting precedent for further secessionist states.

As a note from the moderators: Yes, the premise of the question title is flawed in that there were several border changes in Africa, some of them as recent as the independence of South Sudan. Which is why I assume OP asked "Is this true?".

As a general rule, corrections of the premise of a question unless completely, fundamentally flawed, need to follow the same rules as other answers, namely be in-depth and comprehensive. So when writing up something about border changes in Africa, please be in-depth about it and not just list examples.

Thank you!