This page claims that the longest DTM in this type of endgame is 182 moves.
8/4k3/8/Q7/8/7P/P7/q1K5 w - - 0 1
1. Kc2 Qd4 2. Qe1+ Kf7 3. Qf1+ Kg7 4. Qg2+ Kh8 5. Qa8+ Kh7 6. Qb7+ Kh6 7. Qc6+ Kh5 8. Qe8+ Kg5 9. Qg8+ Kh6 10. Qe6+ Kh7 11. Kb3 Qd3+ 12. Kb4 Qd2+ 13. Kb5 Qd3+ 14. Qc4 Qf5+ 15. Kb6 Qf2+ 16. Kb7 Qf3+ 17. Kc7 Qa3 18. Kb6 Kh8 19. Qd4+ Kh7 20. Qd7+ Kg6 21. Qe6+ Kh5 22. Qd5+ Kg6 23. Kb5 Qb2+ 24. Kc6 Qc3+ 25. Kd7 Qg7+ 26. Kd6 Qf8+ 27. Kc7 Qf2 28. Kc6 Qe2 29. Kd7 Qf2 30. Qe6+ Kg7 31. Kc6 Qf3+ 32. Kb6 Qa3 33. Qb3 Qd6+ 34. Ka7 Qc5+ 35. Kb7 Qe7+ 36. Ka8 Qd8+ 37. Qb8 Qh4 38. Qb7+ Kf8 39. Qc8+ Ke7 40. Kb8 Qa4 41. Qc5+ Kd8 42. Qg5+ Ke8 43. Qe3+ Kf8 44. Qf3+ Ke7 45. a3 Qc4 46. Ka7 Qd4+ 47. Kb7 Qd7+ 48. Kb6 Qd4+ 49. Kb5 Qb2+ 50. Kc5 Qc2+ 51. Kd5 Qd2+ 52. Ke4 Qc2+ 53. Kf4 Qc7+ 54. Ke3 Qa7+ 55. Ke2 Qa6+ 56. Kf2 Qd6 57. Kf1 Qd4 58. Ke2 Qa4 59. Qc3 Ke8 60. Kf3 Qh4 61. Qc8+ Ke7 62. Qg4 Qf6+ 63. Qf4 Qa1 64. Qe3+ Kf8 65. Kg2 Qb1 66. Qf4+ Kg8 67. Qc4+ Kh8 68. Qd4+ Kg8 69. a4 Qg6+ 70. Kf2 Qf5+ 71. Kg3 Qg6+ 72. Kf4 Qf7+ 73. Kg5 Qe7+ 74. Qf6 Qd7 75. Qg6+ Kh8 76. Qh5+ Kg7 77. Qg4 Qd5+ 78. Qf5 Qd8+ 79. Kf4 Qd2+ 80. Kg3 Qe3+ 81. Qf3 Qg1+ 82. Kf4 Qc1+ 83. Kg4 Qc4+ 84. Qf4 Qe6+ 85. Kf3 Qb3+ 86. Kf2 Qb6+ 87. Kg3 Qg6+ 88. Kh4 Qh7+ 89. Kg4 Qg6+ 90. Qg5 Kh8 91. Kf4 Qf7+ 92. Qf5 Qc4+ 93. Qe4 Qc1+ 94. Kf5 Qc8+ 95. Qe6 Qc2+ 96. Kf6 Qf2+ 97. Ke7 Qc5+ 98. Qd6 Qa7+ 99. Qd7 Qc5+ 100. Ke6 Qc4+ 101. Ke5 Qc5+ 102. Ke4 Qc2+ 103. Kd5 Qd1+ 104. Ke6 Qe1+ 105. Kd6 Qb4+ 106. Kc7 Qf4+ 107. Kb6 Qb8+ 108. Kc5 Qe5+ 109. Qd5 Qe7+ 110. Kb6 Qb4+ 111. Qb5 Qd4+ 112. Kb7 Qe4+ 113. Kc8 Qe6+ 114. Kb8 Qg8+ 115. Ka7 Qg7+ 116. Ka8 Qg2+ 117. Qb7 Qxh3 118. a5 Qc3 119. a6 Qc4 120. Qb2+ Kh7 121. Qb6 Qc8+ 122. Ka7 Qc4 123. Kb7 Qe4+ 124. Qc6 Qb4+ 125. Ka7 Qf4 126. Qc8 Qg3 127. Ka8 Qd6 128. Qc2+ Kh6 129. Kb7 Qb4+ 130. Kc6 Qa3 131. Kb5 Kg7 132. Qc7+ Kh8 133. Qc8+ Kh7 134. Qd7+ Kg8 135. a7 Qb3+ 136. Ka6 Qa3+ 137. Kb7 Qb4+ 138. Kc7 Qc4+ 139. Kd8 Qh4+ 140. Qe7 Qd4+ 141. Kc8 Qc4+ 142. Kb8 Qb5+ 143. Qb7 Qe5+ 144. Ka8 Qc5 145. Qb3+ Kf8 146. Kb7 Qe7+ 147. Ka6 Qd6+ 148. Qb6 Qd3+ 149. Kb7 Qd7+ 150. Ka8 Qd5+ 151. Kb8 Qe5+ 152. Kb7 Qe4+ 153. Qc6 Qe7+ 154. Kb8 Qe5+ 155. Ka8 Kg8 156. Qc4+ Kh7 157. Qd3+ Kh8 158. Qd8+ Kh7 159. Kb7 Qb5+ 160. Qb6 Qd5+ 161. Qc6 Qb3+ 162. Kc7 Qg3+ 163. Qd6 Qc3+ 164. Kd7 Qf3 165. Kc8 Qe4 166. Kb8 Qb1+ 167 Kc7 Qc1+ 168. Kd7 Qh1 169. Qc6 Qh3+ 170. Kd8 Qh4+ 171. Kc8 Qb4 172. Qb7+ Qxb7+ 173. Kxb7 Kg7 174. a8=Q Kg6 175. Kc7 Kf5 176. Qa4 Ke5 177. Qg4 Kd5 178. Qf4 Kc5 179. Qe4 Kb5 180. Qd4 Ka5 181. Kc6 Ka6 182. Qb6#
According to the Sygyzy tablebases, the longest DTZ is 171 moves.
8/K6k/8/8/P7/Q6P/8/1q6 b - - 0 1
Records for DTC are not provided and I was not able to find any. DTC in this case occurs at 173, which is pretty high and a good estimate for a possible record holder (given the maximum DTM is at 182).