Following Szabolcs adviceSzabolcs advice, it seems that the Documentation page for EvaluationMonitor contains all what you need for Method -> "Newton":
data = {{0, 1}, {1, 0}, {3, 2}, {5, 4}, {6, 4}, {7, 5}};
Clear[evalCount];
evalCount[_] = 0;
nlm = NonlinearModelFit[data, Log[a + b x^2], {a, b}, x,
EvaluationMonitor :> ++evalCount["Function"],
Gradient -> {"Symbolic", EvaluationMonitor :> ++evalCount["Gradient"]},
Method -> {"Newton",
"Hessian" -> {"Symbolic", EvaluationMonitor :> ++evalCount["Hessian"]}}];
TableForm[evalCount /@ #, TableHeadings -> {#, None}] &@{"Function", "Gradient",
"Hessian"}
Other Methods take different suboptions, for example "LevenbergMarquardt":
Clear[evalCount];
evalCount[_] = 0;
nlm = NonlinearModelFit[data, Log[a + b x^2], {a, b}, x,
EvaluationMonitor :> ++evalCount["Function"],
Gradient -> {"Symbolic", EvaluationMonitor :> ++evalCount["Gradient"]},
Method -> {"LevenbergMarquardt",
"Residual" -> {"Symbolic", EvaluationMonitor :> ++evalCount["Residual"]},
"Jacobian" -> {"Symbolic", EvaluationMonitor :> ++evalCount["Jacobian"]}}];
TableForm[evalCount /@ #, TableHeadings -> {#, None}] &@{"Function", "Gradient",
"Residual", "Jacobian"}
Although I'm not sure why inside of Method suboptions EvaluationMonitor does not catch anything for NonlinearModelFit...
