Run the application

import com.sun.javafx.runtime.VersionInfo;
import javafx.application.Application;
import javafx.beans.property.SimpleStringProperty;
import javafx.beans.value.ObservableValue;
import javafx.event.ActionEvent;
import javafx.event.EventHandler;
import javafx.scene.Scene;
import javafx.scene.control.Button;
import javafx.scene.control.TreeItem;
import javafx.scene.control.TreeTableColumn;
import javafx.scene.control.TreeTableView;
import javafx.scene.layout.HBox;
import javafx.scene.layout.VBox;
import javafx.stage.Stage;
import javafx.util.Callback;

public class TreeTableSample1 extends Application {

    TreeTableView<Person> treeTableView;
    TreeTableColumn<Person, String> firstNameCol;

    @Override
    public void start(Stage primaryStage) {

        initTreeTableView();

        HBox root = new HBox(10.0);
        root.getChildren().add(treeTableView);

        Button action = new Button("Increase min width");
        action.setOnAction(new EventHandler<ActionEvent>() {

            @Override
            public void handle(ActionEvent actionEvent) {
                System.out.println("Setting min width to 500");
                firstNameCol.setMinWidth(500);
            }
        });

        VBox vb = new VBox(10.0);
        vb.getChildren().add(action);

        root.getChildren().add(vb);

        Scene scene = new Scene(root, 400, 300);

        primaryStage.setTitle(VersionInfo.getRuntimeVersion());
        primaryStage.setScene(scene);
        primaryStage.show();
    }

    public static void main(String[] args) {
        launch(args);
    }

    private void initTreeTableView() {
        treeTableView = new TreeTableView<Person>();
        treeTableView.setMinSize(200, 150);
        treeTableView.setPrefSize(200, 200);
        treeTableView.setMaxSize(200, 250);
        treeTableView.setColumnResizePolicy(TreeTableView.CONSTRAINED_RESIZE_POLICY);

        TreeItem<Person> root = new TreeItem<Person>(new Person("Anna"));
        root.setExpanded(true);
        treeTableView.setRoot(root);
        treeTableView.showRootProperty().set(true);

        firstNameCol = new TreeTableColumn<Person, String>("First name");
        firstNameCol.setCellValueFactory(new Callback<TreeTableColumn.CellDataFeatures<Person, String>, ObservableValue<String>>() {
            @Override
            public ObservableValue<String> call(TreeTableColumn.CellDataFeatures<Person, String> p) {
                return p.getValue().getValue().firstName;
            }
        });
        firstNameCol.setMinWidth(80d);

        treeTableView.getColumns().addAll(firstNameCol);

        Person p = new Person("Bob");
        treeTableView.getRoot().getChildren().add(new TreeItem<Person>(p));

        p = new Person("Cindy");
        treeTableView.getRoot().getChildren().add(new TreeItem<Person>(p));

        p = new Person("Zack");
        treeTableView.getRoot().getChildren().add(new TreeItem<Person>(p));
    }

    class Person implements Comparable{

        public final SimpleStringProperty firstName;

        public Person(String name) {
            this.firstName = new SimpleStringProperty(name);
        }

        public String toString() {
            return firstName.get();
        }

        @Override
        public int compareTo(Object o) {
            return this.firstName.get().compareTo(((Person) o).firstName.get());
        }
    }
}

Click the button.
Scroll to the right end.
Move mouse to the right side of the column header.
You will see mouse looking like "<->" which denotes, that you can resize something. But you cannot resize this column, or anything else.