Output Matrix Content from Sibling
Something like this might work (obviously, replace the matrixField, assetsField and imageTransform handles with real values):

{# Basic query parameters for your sibling entries #}
{% set siblingCriteria = craft.entries.section(entry.section.handle) %}

{# Creates an array with the next and previous entries, based on the basic query parameters #}
{% set siblingEntries = [entry.getPrev(siblingCriteria), entry.getNext(siblingCriteria)] %}

{# Loop through the next and previous entries
{% for siblingEntry in siblingEntries %}

    {% if siblingEntry %}{# This might be null, if there isn't a next or previous entry #}

        {# Get the first `gallery` block for the entry's Matrix field #}
        {% set galleryBlock = siblingEntry.matrixField.type('gallery').first() %}

        {# Get the first asset in the gallery block's Assets field #}
        {% set galleryImage = galleryBlock ? galleryBlock.assetsField.first() : null %}

        {# Render the image #}
        {% if galleryImage %}
            &lt;a href="{{ siblingEntry.url }}"&gt;
                &lt;img src="{{ galleryImage.getUrl('imageTransform') }}" /&gt;
            &lt;/a&gt;
        {% endif %}

    {% endif %}

{% endfor %}