Community search, which finds cohesive subgraphs containing given query vertices, has attracted much attention in decades. On attributed graphs, when considering the fairness of members' attributes in a community, the cohesiveness constraint of a clique is too strong, which often causes no fair clique based communities can be found. Thus, in this paper, we use the k-truss model, which is a relaxation of the clique but whose members have large engagement and high tie strength, to describe fair communities, namely fair k-truss communities (FTC) and anchored fair k-truss communities (AFTC, using anchored vertices to help satisfying the fairness constraint). We formulate the FTC and AFTC search problems to find the FTC or AFTC containing a given query vertex$q$which has the largest$k$and the smallest diameter. We prove the hardness of both problems. We develop several greedy algorithms and acceleration strategies to solve FTC and AFTC search problems. Experiments on 8 real-world networks show the significance of our FTC and AFTC models, and high performance of our algorithms and acceleration strategies.