How to set TextInputLayout error message colour?

Actually, to change just the error message color, you can set textColorError in your theme (and also set colorControlNormal and colorControlActivated for the general widget and hint text color). TextInputLayout picks up that attribute. NOTE: if you set errorTextAppearance to a custom style then textColorError will have no effect.

<style name="AppTheme" parent="Theme.AppCompat.Light.DarkActionBar">
    <item name="colorControlNormal">@color/control_normal</item>
    <item name="colorControlActivated">@color/control_activated</item>
    <item name="textColorError">@color/error</item>
    <!-- other styles... -->
</style>

And in your AndroidManifest.xml:

<application
    android:theme="@style/AppTheme"
    android:icon="@drawable/ic_launcher"
    android:label="@string/app_name">

    <!-- ... -->

</application>

With the TextInputLayout included in the Material Components Library just use the app:errorTextColor attribute.

    <com.google.android.material.textfield.TextInputLayout
        app:errorTextColor="@color/...."
        .../>

In a custom style you can use:

<style name="..." parent="Widget.MaterialComponents.TextInputLayout.FilledBox" >
   <item name="errorTextColor">@color/...</item>
   ...
</style>

One side note. I have tried the accepted solution one with errorTextAppereance. It works really good but at first, the input underline color was not changing after applying a new errorTextAppereance style. I see there are a few comments and that other people are experiencing the same issue.

In my case, this was happening when I was setting a new style after setting a new error text. Like this:

passwordInputLayout.error = "Password strength"
passwordInputLayout.setErrorTextAppearance(R.style.InputError_Purple)

After switching the order of this two methods the text and underline color changes as expected.

passwordInputLayout.setErrorTextAppearance(R.style.InputError_Purple)
passwordInputLayout.error = "Password strength"

And the error text appearance style looks something like this:

<style name="InputError" parent="TextAppearance.Design.Error"/>
<style name="InputError.Purple">
    <item name="android:textColor">@color/purple</item>
</style>

Create a custom style which uses @android:style/TextAppearance as parent in your styles.xml file:

<style name="error_appearance" parent="@android:style/TextAppearance">
    <item name="android:textColor">@color/red_500</item>
    <item name="android:textSize">12sp</item>
</style>

And use it in your TextInputLayout widget:

 <android.support.design.widget.TextInputLayout
            android:id="@+id/emailInputLayout"
            android:layout_width="match_parent"
            android:layout_height="wrap_content"
            app:errorTextAppearance="@style/error_appearance">

Edit: Set the hint on the object, which is inside your TextInputLayout (EditText, TextView, etc.) to hold different colors for the hint and the error.