Yes, the Z-values used in Logistic regression do come from the Wald test.
While Minitab does not state this, you can look up the formula for Z in the Logistic regression help menu (Z = coefficient/SE) and compare it to the formula for the Wald test. They are identical. In addition, Minitab does include a reference to a book on the Wald test.
"Goodness-of-Fit Tests - displays Pearson, deviance, and Hosmer-Lemeshow goodness-of-fit tests. In addition, two Brown tests-general alternative and symmetric alternative-are displayed because you have chosen the logit link function and the selected option in the Results subdialog box. The goodness-of-fit tests, with p-values ranging from 0.312 to 0.724, indicate that there is insufficient evidence to claim that the model does not fit the data adequately. If the p-value is less than your accepted a-level, the test would reject the null hypothesis of an adequate fit."
Therefore your p-value is indicative of a good model fit.
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